This work aims for a new stochastic resonance (SR) model which performs well in bearing fault diagnosis. Different from the traditional bistable SR system, we realize the SR based on the joint of Woods-Saxon potential (WSP) and Gaussian potential (GP) instead of a reflection-symmetric quartic potential. With this potential model, all the parameters in the Woods-Saxon and Gaussian SR (WSGSR) system are not coupled when compared to the traditional one, so the output signal-to-noise ratio (SNR) can be optimized much more easily by tuning the system parameters. Besides, a smoother potential bottom and steeper potential wall lead to a stable particle motion within each potential well and avoid the unexpected noise. Different from the SR with only WSP which is a monostable system, we improve it into a bistable one as a general form offering a higher SNR and a wider bandwidth. Finally, the proposed model is verified to be outstanding in weak signal detection for bearing fault diagnosis and the strategy offers us a more effective and feasible diagnosis conclusion.

Stochastic resonance (SR) has been firstly introduced by Benzi and coworkers [

On the other hand, rotating machinery plays a significant role in a wide range of industrial applications, such as transportation vehicles, aeroengine, and power generators [

Noise filtering seems a common and effective method which can suppress the noise and improve the output signal-to-noise ratio (SNR), while the SR offers us a new approach and idea. As compared to traditional techniques that mainly focus on how to suppress the noise, the SR has the merit of signal enhancement by the aid of the noise which means the noise plays an active role in our work [

In this paper, we change the WSSR from monostable into bistable by the cooperation of a Gaussian potential (GP). The WSP was firstly put into use by Deza et al. for wide spectrum energy harvesting based on the SR principle [

The rest of the paper is arranged as follows. In Section

In WSP model, the potential function

Here parameter

Shape of WSP

The attractive radial Gaussian potential of the form

Shape of GP

With the two introduced models, we proposed a new bistable potential model WSG by the combination of WSP and GP which can be expressed as follows:

Its potential distribution is shown in Figure

Shape of WSGP

The three basic ingredients of producing SR phenomenon are

In (

Equation (

In fact, the effect of SR is the result of the joint action of the system, input signal, and random noise. We build a specific scene that the particle oscillates within the potential under the collective excitation from the potential force, the periodic force, and the noise force to describe SR phenomenon concretely. Among all the kinds of force, the potential force is generated by the gradient of the potential curve which can be described as the first-order derivative of

Generally speaking, for the shape of the driving potential such as (

With these features, we can find the advantages of the WSGSR over the traditional one qualitatively. In order to make the potential model in an optimal status, we need to adjust the parameters. For the traditional potential model as in (

To optimally detect the weak signal of driving frequency from the background noise, the WSGSR can be conducted by adjusting the WSGP parameters until the optimal SR output signal which we get with the method mentioned above is achieved. While this is not enough, we need a target to evaluate the performance of SR effect. Here, we employ the output SNR as a criterion to assess the result with which we can judge the optimal condition of our SR system. The output SNR is defined as the power spectral density of the driving signal divided by the average background noise in a small frequency bin around the driving frequency [

The SR model is shown as in (

Then based on the new WSGSR model, we propose a new scheme of bearing fault diagnosis with this SR system, which is presented in Figure

Proposed scheme of bearing fault diagnosis with WSGSR.

To have a general presentation for the effect of the proposed SR model, we simulate a sinusoidal signal with frequency of

Firstly, we take

Outputs signal of different SR models with a sinusoidal signal mixed with noise: (a) sinusoidal signal, (b) original signal mixed with noise, (c) traditional bistable potential with

As the power spectral density of SR output meets the Lorentzian distribution which is characterized by concentrating most of noise energy into the low frequency region. So there is a small parameter limitation for SR effect [

In practical bearing fault diagnosis, bearing fault signal is always present in the form of impactive series with a fault frequency modulated to a high level [

Here mod

Simulated bearing fault signal and outputs of different models: (a) periodic unilateral attenuation impulse signal of 100 Hz and mixed signal with noise intensity of 1.8, (b) envelope signal and power spectrum filtered with

Sending the above signal demodulated by HT into the traditional bistable SR model, we get the output signal and its power spectrum as in Figure

The outputs we get in Figure

Before we search for the optimal output, we need to have the signal preprocessed. As the original signal is modulated, we firstly need to have it band-pass filtered. Then use Hilbert transform to process the signal. The output will be demodulated successfully. After this, we can send the signal into WSGSR system of different parameters to gain the output.

Calculate the power spectrum of the output waveform and obtain the SNR. Search the maximal SNR in the parameter space that is constructed by the varying variables

Finally, substitute the optimal parameters (as we get in Figure

Optimal result for Figure

In the optimization process above, we mentioned that optimal parameters can be obtained according to the maximal SNR in a certain range. It means that there will appear a peak for SNR with only one of the parameters changing. With a decided input signal, the system output only depends on five potential parameters. We fixed four of them and varied the other one of the WSG potential in turn and then calculated the SNR of the output signal via (

SNRs variation trend with the changing of five parameters in different ranges: (a)

Following on, we make a new simulated signal with a higher driving frequency of

Outputs of different models with driving frequency of 200 Hz: (a) original waveform and its power spectrum with

In order to verify our conjecture about the proposed method, we make a simulation to compare the capacity of different models when dealing with signal under different noise intensity or driving frequency.

Firstly, fixing the frequency at 200 Hz still, we set the amplitude of attenuation impulse signal in (

Output SNRs under different noise and driving frequency levels: (a) output SNR with the changing of noise intensity and (b) output SNR with the changing of driving frequency.

Secondly, to observe the performances of different models under different driving frequency, we make another series signal. In them, the amplitude of attenuation impulse signal is set as

Based on the above analysis, our conjecture is verified. With the proposed WSGSR model, signal extract under heavy noise or bearing fault diagnosis can be obtained by tuning the independent parameters and it is proved to have a better adaptability in detecting the weak signal with different noise intensities and different driving frequencies. After these simulation works, the model will be applied in experimental bearing fault signal to value its engineering application.

To verify the effectiveness and efficiency of the proposed method in engineering applications, a set of train bearing signals carrying fault information are analyzed according to the scheme of bearing fault diagnosis with proposed WSGSR model in Figure

The train bearings used in this experiment are single row radial short cylindrical roller bearings, with the type of NJ(P)3226X1 with its detailed specification in Table

Specification of the train bearing NJ(P)3226X1.

Type | Diameter of the outer race | Diameter of the inner race | Pitch diameter ( |
Diameter of the roller ( |
Number of the roller ( |
---|---|---|---|---|---|

NJ(P)3226X1 | 250 mm | 130 mm | 190 mm | 32 mm | 14 |

Artificial cracks on the train bearing: (a) outer-race crack and (b) inner-race crack.

Experiment platform for bearing fault signal acquisition.

With all the introduced equipment, we can get the expecting bearing fault signal. In the experiment, the rotating speed is set at 1430 r/min, with a load of about 3 t and the sampling rate of 50 kHz. Through the calculation of bearing fault frequency equation with the parameters in Table

The analyzed results of the outer-race defective train bearing signal using different methods are displayed in Figure

Optimal outputs of different models with outer-race fault signal: (a) original waveform and its power spectrum, (b) envelope signal and its spectrum filtered with [500 Hz and 2500 Hz], (c) output of bistable SR model and its spectrum with

The analyzed results of the inner-race defective train bearing signal dealing with different systems are displayed in Figure

Optimal outputs of different models with inner-race fault signal: (a) original waveform and its power spectrum, (b) envelope signal and its spectrum filtered with [500 Hz and 2500 Hz], (c) output of bistable SR model and its spectrum with

With the above subsections, the effectiveness and efficiency of the proposed weak signal detection strategy based on WSGSR have been verified by both the simulated signal and the practical bearing fault signals. The results show a similar verdict.

Considering the two expressions of different SR models as in (

Compared to the shape of traditional bistable potential, the WSGD has a flatter well bottom and steeper well wall. This might make the particle oscillate more easily with less resistance and higher rebound force which we have described before. The outputs verify our conjecture that the SR effect comes up more easily in the proposed model. The simulating and engineering results all provide the evidence that WSGSR results in a higher output SNR. More commendable, it can work with a wider range of noise intensity and driving frequency. In other words, even with heavier noise and higher frequency, the advantages of proposed method are more obvious and it can offer a better performance than the traditional one. This will owe to the special shape of WSGP.

Besides, as a more general type, the WSGSR not only has the advantages of WSSR which is a particular case but also possesses a better performance. With utilizing of a potential barrier which can be turned by

We have mentioned that when considering the amplitude of output spectrum in Figures

In engineering applications, the key is how to determine the model parameters. However, for an unknown system, we cannot obtain the fault samples in advance. In this case, we know in most of the practical applications that the fault information cannot be obtained easily before the diagnosis work. When processing the signal with band-pass filter and HT, we might not distinguish the fault frequency effectively due to heavy background noise. Then with just a SR model whose parameters are preset, we can extract the component which might not be so distinct. But with this information, we can use the component to adjust the parameters to the optimal combination according to the highest SNR which will make the fault frequency the clearest. If the component is exactly the periodic fault signal, it can be amplified to a high level by the optimal parameters but not vice versa. So the proposed method can offer us a more effective approach to make an accurate and reliable diagnosis.

However, the response of the WSGSR system is still complex and sometimes comes up with randomness as the acquired signal and noise always contain the uncertainty. Hence, a small variation of parameters may result in a quite different output like butterfly effect. What is more is that the model needs some mathematical analysis to support the certain effects that parameters’ work has on the output. A deeper mechanism of the WSGSR from both the mathematical and the physical aspects should be investigated. These further studies will make the parameter selection and optimization more effective and gain a more satisfying output.

An improved potential model of WSGP is investigated to realize the SR effect instead of the traditional bistable one. The model of WSGSR with a particular form of WSSR

The authors declare that there is no conflict of interests regarding the publication of this paper.

This work was supported in part by the National Natural Science Foundation of China with Grant nos. 51075379 and 11274300. The authors also would like to thank the anonymous reviewers for their valuable comments and suggestions.